Match List I with List II
| LIST I | LIST II | ||
| A. | The solution set of the inequality \(3x + 7 \gt 12\) | I. | \([-1, \infty]\) |
| B. | The solution set of the inequality \(\displaystyle\frac{3x + 5}{2} \ge 1, x \in \mathrm{R}\) | II. | \(\left[\displaystyle\frac{17}{8}, \infty\right]\) |
| C. | The solution set of the inequality \(2x + 5 \lt 7x + 9\), \(x \in \mathrm{R}\) is, | III. | \(\left(\displaystyle\frac{5}{3}, \infty\right)\) |
| D. | The solution set of the inequality \(6x - 5 \ge -2x + 12\), \(x \in \mathrm{R}\) is, | IV. | \(\left(-\displaystyle\frac{4}{5}, \infty\right)\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A-III, B-I, C-IV, D-II
Each inequality is solved for x. A strict inequality (< or >) gives an open end ( ), while ≤ or ≥ gives a closed end [ ]. The end at \(\infty\) is always open.
A. \(3x + 7 \gt 12\)
This matches III. \(\left(\frac{5}{3}, \infty\right)\).
\(3x \gt 5\)
\( \displaystyle x \gt \frac{5}{3} \)
B. \(\frac{3x + 5}{2} \ge 1\)
This matches I. \([-1, \infty]\), that is, all x ≥ −1.
\(3x + 5 \ge 2\)
\(3x \ge -3\)
\(x \ge -1\)
C. \(2x + 5 \lt 7x + 9\)
This matches IV. \(\left(-\frac{4}{5}, \infty\right)\).
\(-4 \lt 5x\)
\( \displaystyle x \gt -\frac{4}{5} \)
D. \(6x - 5 \ge -2x + 12\)
This matches II. \(\left[\frac{17}{8}, \infty\right]\), that is, all \(x \ge \frac{17}{8}\).
\(8x \ge 17\)
\( \displaystyle x \ge \frac{17}{8} \)